Sub-Riemannian geometry working seminar, organizational matters
– We meet every wednesday, 5pm, at IHP, amphi Darboux (except for a few exceptional days where we move to room 05, same floor). If people prefer a different schedule, let me know.
– People welcome to join for dinner after seminar.
– Many people have offered to give a talk. After today’s talk, we shall sort out proposals.
1. Metric spaces not parametrized by their tangents
1.1. Sub-Finsler manifolds
Definition 1 An (equiregular) sub-Finsler manifold is the data of a Finsler manifold (a smoothly varying norm is given on each tangent space) and a smooth sub-bundle satisfying the following assumptions. Let denote the family of vector fields obtained iteratively from smooth sections of by the Lie bracket operation,
Let denote the set of values at point of elements of . Then we assume that
- There exists such that .
- All are sub-bundles.
A sub-Finsler manifold comes up with a distance: given two points, minimize length of curves between them which are almost everywhere tangent to . Note this distance only depends on the restriction of the norm to . So in fact one does not need the norm is other directions. It is a leitmotiv of sub-Finsler geometry: although, occasionnally, we may use geometric data defined outside , what really matters for us is and the associated metric.
1.2. Carnot groups
Carnot groups form an important class of examples.
Definition 2 Let be a Lie algebra. Say that a direct sum decomposition is a stratification of if
Pick a norm on , left-translate it, this gives a sub-Finsler manifold.
Example 1 with , and is the determinant. This is called the first Heisenberg Lie algebra.
1.3. Tangent cones
Theorem 3 (Mitchell 1985, completed by other people) The tangent cone of a sub-Finsler manifold et a point is a Carnot group (which may depend on ).
Here, tangent cone of at means the limit of dilates of . Limit is taken in Gromov-Hausdorff sense.
Definition 4 Let , be bounded metric spaces. Define
where inf is taken over all metric spaces , and all subsets isometric to and isometric to , and maps preserve base points.
For unbounded pointed metrc spaces and , say that tends to if for all , the -balls converge, i.e.
Here we are in fact dealing with (pointed) metric spaces up to isometry.
Metric spaces need not admit tangent cones. The doubling condition is a favorable circumstance: it guarantees compactness of dilates, and so existence of converging subsequences among dilates. Nevertheless, limits need not exist (i.e. sub-limits need not be unique), even in the class of doubling metric spaces.
Theorem 5 (Le Donne) Let be a geodesic metric space. Assume has a doubling measure . Assume that -almost everywhere, has only one tangent cone (i.e. dilates converge to some metric space). Then the tangent cone is a Carnot group -almost everywhere.
The proof uses the following
Lemma 6 In a doubling metric space , almost every point has the following property. Assume is a tangent cone to at . Then for all , is again a tangent cone of at .
and relies essentially on the following
Theorem 7 (Gleason, Montgomery, Zippin 1950, Berestovskii 1988) Assume is a complete, proper, connected and locally connected metric space. Assume isometry group is transitive. Then is a Lie group with finitely many connected components.
Moreover (Berestovskii), if is geodesic, then is a sub-Finsler manifold.
1.4. Impossible parametrization by tangents
For sub-Riemannian manifolds, the tangent space is not a local model in general.
Theorem 8 (Le Donne, Ottazzi, Warhurst 2011) There exists a nilpotent Lie group equipped with a left-invariant sub-Riemannian metric which is not locally quasiconformal to its tangent cones.
Note that bi-Lipschitz quasiconformal.
Definition 9 Say a Carnot group is ultra rigid if every even locally defined quasiconformal self-mapping is the restriction of the composition of a translation and a dilation.
Theorem 8 easily follows from the following
Lemma 10 There exists a nilpotent non Carnot Lie group equipped with a left-invariant sub-Riemannian distance for which the tangent cone is ultra rigid.
Indeed, ultra rigidity provides an injective group homomorphism from to the group of translations/dilations of . The semi-direct product is not nilpotent, so homomorphism is not onto (at the level of Lie algebras), maps injectively to . Since they have the same dimension, , contradiction.